tri-ellipse_construction.ggb |
A circle is the collection of all points that are a fixed distance from a focus (center). An ellipse is the collection of points, the sum of whose distances to two fixed foci is constant. These shapes are covered extensively in many high school classes. A tri-ellipse is the collection of points, the sum of whose distances to three fixed foci is constant. As we have seen, these shapes can take some unfamiliar forms . What else can we understand about these shapes?
It is possible to use geometric software to perform an asymmetric construction of a tri-ellipse. Let the foci of the tri-ellipse be points $A, B, \text{ and } C$ and the distance-sum be $d$. Write $d$ as $d=e+(d-e)$ for any $0\leq e \leq d.$ Draw the ellipse with foci $A \text{ and } B$ and distance-sum $e.$ Draw the circle with center $C$ and radius $d-e.$ The intersection(s) of the ellipse with the circle lie on the tri-ellipse. The tri-ellipse is then the locus of such intersection points as $e$ varies from $0$ to $d.$ Experiment with the applet below to create your own tri-ellipses.
You can also download the Geogebra file here if you want to look into the nuts and bolts.
Additionally, the tri-ellipse can be approached from an algebraic perspective. If the foci $A, B, \text{ and } C$ have coordinates $(x_A, y_A), (x_B, y_B), \text{ and } (x_C, y_C),$ then the tri-ellipse is represented by the equation
$$\sum_{L=A, B, C} \sqrt{(x-x_L)^2+(y-y_L)^2}=d.$$
Squaring three times, combined with some algebraic manipulations turns this equation into a degree-eight polynomial in two variables. Perhaps there are some algebraic techniques to bring to bear to further elucidate properties of the tri-ellipse.
Open Questions
There is a parametrization of an ellipse given by $x = r \cos \theta\text{ and } y = r\sin\theta.$ Is there a similar parametrization for a tri-ellipse in terms of an angle? This parametrization would be centered at the point found in the last post.
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I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Hanoi, Vietnam. Your comments are always welcome. Archives
May 2021
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